scholarly journals An Extension of Hypercyclicity forN-Linear Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Juan Bès ◽  
J. Alberto Conejero
Keyword(s):  
Difference Equations ◽  
Entire Functions ◽  
Linear Operators ◽  
Fréchet Space ◽  
Infinite Dimensional ◽  
Bilinear Operators ◽  
Hypercyclic Vectors ◽  
Countable Product ◽  
Dense Orbits ◽  
Residual Sets

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit forN-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclicN-linear operators, for eachN≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines supportN-linear operators with residual sets of hypercyclic vectors, forN=2.

Central Limit Theorem in View of Subspace Convex-Cyclic Operators

2021 ◽  
Vol 103 (3) ◽  
pp. 25-35
Author(s):  
H.M. Hasan ◽  
◽  
D.F. Ahmed ◽  
M.F. Hama ◽  
K.H.F. Jwamer ◽  
...  
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Ergodic Theory ◽  
Linear Operators ◽  
Cyclic Operator ◽  
Infinite Dimensional ◽  
System Operator ◽  
Cyclic Operators ◽  
Dense Orbits

In our work we have defined an operator called subspace convex-cyclic operator. The property of this newly defined operator relates eigenvalues which have eigenvectors of modulus one with kernels of the operator. We have also illustrated the effect of the subspace convex-cyclic operator when we let it function in linear dynamics and joining it with functional analysis. The work is done on infinite dimensional spaces which may make linear operators have dense orbits. Its property of measure preserving puts together probability space with measurable dynamics and widens the subject to ergodic theory. We have also applied Birkhoff’s Ergodic Theorem to give a modified version of subspace convex-cyclic operator. To work on a separable infinite Hilbert space, it is important to have Gaussian invariant measure from which we use eigenvectors of modulus one to get what we need to have. One of the important results that we have got from this paper is the study of Central Limit Theorem. We have shown that providing Gaussian measure, Central Limit Theorem holds under the certain conditions that are given to the defined operator. In general our work is theoretically new and is combining three basic concepts dynamical system, operator theory and ergodic theory under the measure and statistics theory.

scholarly journals On Spectral Characterization of Nonuniform Hyperbolicity

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Muna Abu Alhalawa ◽  
Davor Dragičević
Keyword(s):  
Linear Operators ◽  
Sharp Contrast ◽  
Spectral Characterization ◽  
Fréchet Space ◽  
Nonuniform Hyperbolicity ◽  
Exponential Dichotomies ◽  
Infinite Dimensional ◽  
Theoretic Characterization ◽  
Principal Advantage

We give a complete functional theoretic characterization of tempered exponential dichotomies in terms of the invertibility of certain linear operators acting on a suitable Frechét space. In sharp contrast to previous results, we consider noninvertible linear cocycles acting on infinite-dimensional spaces. The principal advantage of our results is that they avoid the use of Lyapunov norms.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

Elements of Spectral Theory for Generalized Derivations II : The Semifredholm Domain

1981 ◽  
Vol 33 (5) ◽  
pp. 1205-1231 ◽  
Author(s):  
Lawrence A. Fialkow
Keyword(s):  
Hilbert Spaces ◽  
Simple Formula ◽  
Decomposition Theorem ◽  
Linear Operators ◽  
Generalized Derivations ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Hilbert Space Operators ◽  
Algebraic Invariants ◽  
Fredholm Theorem

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

scholarly journals Necessary conditions of optimality for infinite dimensional uncertain systems

1995 ◽  
Vol 1 (3) ◽  
pp. 179-191 ◽  
Author(s):  
N. U. Ahmed ◽  
X. Xiang
Keyword(s):  
Uncertain Systems ◽  
Infinitesimal Generator ◽  
Continuous Semigroup ◽  
Computational Algorithm ◽  
Necessary Conditions ◽  
Reflexive Banach Space ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Necessary Conditions Of Optimality ◽  
Semigroup Of Linear Operators

In this paper we consider optimal control problem for infinite dimensional uncertain systems. Necessary conditions of optimality are presented under the assumption that the principal operator is the infinitesimal generator of a strongly continuous semigroup of linear operators in a reflexive Banach space. Further, a computational algorithm suitable for computing the optimal policies is also given.

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